Abstract

A nonlinear equation of motion of vibrating membrane with a “viscosity” term is investigated. Usually, the term is added, and it is well known that this equation is well posed in the space of functions. In this paper, the viscosity term is changed to , and it is proved that if initial data is slightly smooth (but belonging to is sufficient), then a weak solution exists uniquely in the space of BV functions.

1. Introduction

Let be a bounded domain in with the Lipschitz continuous boundary . In [1] and in the author’s previous works [2–4], the following:
is investigated, which is in these works referred to as the equation of motion of vibrating membrane. Up to now, neither existence nor uniqueness of a solution to (1) is obtained. In [1–3], we only have that a sequence of approximate solutions to (1) converges to a function in an appropriate function space, and that if satisfies the energy conservation law, it is a weak solution to (1). In [1], approximate solutions are constructed by the Ritz-Galerkin method and in [2, 3] by Rothe’s method. In [2], the boundary condition is not essentially discussed, and the observation is added in [3]. In these works, the limit should satisfy the energy conservation law, and existence theorem of a global weak solution has not been established yet. Instead, in [4], linear approximation for (1) is established. On the other hand, the equation with the strong viscosity term is investigated by several authors. For example, in [5], it is investigated in the context of control theory, and it is asserted that if and , there exists a unique solution for each . Namely, the equation with strong viscosity term is well posed in , and since is a smaller class than the space of BV functions, this suggests that the influence of the term is too strong.

In this paper, replacing the strong viscosity term with , we investigate it in the space of BV functions. Namely, our problem of this paper is as follows:
with initial and boundary conditions
We should note that the term “viscosity” probably means implying regularity. However, in this paper, we only investigate existence and uniqueness of (2)–(4), regularity is not investigated. This is the reason that in the title there is a quotation mark.

A function is said to be a function of bounded variation or a BV function in if the distributional derivative is an valued finite Radon measure in . The vector space of all functions of bounded variation in is denoted by . It is a Banach space equipped with the norm (see, e.g., [6–8]). We should note that, for , the operator is multivalued. It is usually defined by the use of the subdifferential of the area functional. Namely, for a function , we regard as
where
Here, readers should note that is not . We are imposing (4), and in the analysis in the space of BV functions, the most appropriate weak formulation of (4) is to replace with (cf. [3], see, also [4, Appendix C]).

Definition 1. A function is a weak solution to (2)–(4) in if satisfies that(i), , (ii)s- in , (iii) there exist and such that for -a.e. ( denotes the one-dimensional Lebesgue measure), and, for any ,
If a function is a weak solution to (2)–(4) in for each , then we say that is a weak solution to (2)–(4) in .

Our main theorem is as follows.

Theorem 2. Suppose that and . We further suppose that . Then, there exists a unique weak solution to (2)–(4) in .

Remark 3. If , then the element is unique. Indeed, for each , is differentiable at and for each . Since and is arbitrary, is uniquely determined.

2. Reduction of the Problem

In order to solve (2)–(4), we give a formal observation. Let us put
then (2) becomes , which can be regarded as an ordinary differential equation to . By the variation-of-constants formula, we obtain that . Noting that , we have
where . Hence, formally, (2) is reduced to
Definition of a weak solution to this equation is as follows.

Definition 4. Let , , and . A function is a weak solution to (10) with (3) and (4) in if satisfies that(i), , (ii)s- in , (iii)for any and for -a.e. ,
Similar to the case of (2), we say that is a weak solution to (10) with (3) and (4) in if a function is a weak solution to (10) with (3) and (4) in for each .The previous observation is just formal. In the following proposition, we show it rigorously.

Proof. It is sufficient to show that, for each , a function is a weak solution to (2)–(4) in if and only if it is a weak solution to (10) with (3) and (4) in . Suppose that is a weak solution to (10) with (3) and (4) in . Conditions (i) and (ii) of Definition 1 are the same as those of Definition 4. Thus, we only have to show (iii) of Definition 1. Let
Then, by (iii) of Definition 4, we have that for -a.e. . Thus, by a direct calculation, we have that satisfies (iii) of Definition 1. Next, we suppose that is a weak solution to (2)–(4) in . For each and each , we put
Then , and since , we have the following by (iii) of Definition 1:
By integration by parts, we have
Furthermore, we have the following by Fubini’s theorem:
Finally, noting that , we have the following by (14), (15), and (16):
Since and are arbitrary, we have, for -a.e. ,
which means that satisfies Definition 4, (iii).

Theorem 6. Suppose that and . We further suppose that and let . Then, there exists a unique weak solution to (10), (3), and (4) in .

Our strategy of proving Theorem 6 is the contracting mapping theorem. For this purpose, given that , we solve
and show that the map is a contraction. A weak solution to (19) with (3) and (4) is defined as follows.

Definition 7. Let . A function is a weak solution to (19) with (3) and (4) in if satisfies that (i), , (ii)s- in , (iii)for any and for -a.e. ,
The proof of Theorem 6 consists of two parts. The first part is solving (19), and the second part is to show that the map is a contraction.

Let and be as in Theorem 6. In this section, we show that there exists a unique solution to (19) with (3) and (4) in for each .

Uniqueness is easy. Suppose that and are solutions to (19) with (3) and (4) in , and inserting to (iii) of Definition 7, integrating it from to , obtaining another inequality by replacing and , and adding these two inequalities, we have

It is sufficient to show the existence in for -a.e. . Approximate solutions are constructed by Rothe’s time semidiscretization method. In Rothe’s method, we should solve elliptic equations with respect to space variables. Here, we solve them by a direct variational method (namely, this is the method of discrete Morse semiflow, cf. [9] and references cited therein).

Suppose that with and , and let . For a positive number , we construct a sequence in the following way. For , we let be as in (3), and for , it is defined as a minimizer of the following functional:
in the class , where
Since
is bounded from below, and hence, the existence of a minimizer of follows.

Lemma 8 (energy inequality). consider the following:

Proof. Since is a minimizer of , we have
Hence, for each ,
Thus, by induction on , we have the conclusion.

Next, we define approximate solutions and for as follows: for ,
Then Lemma 8 shows for each

Now, we estimate the second term of the left hand side of (29). Then,
where , and thus, it is easy to see that
By (29), we have, for each and for each ,
where

Proof. Assertion (1) immediately follows from (32). Since we have
for each , , Assertion (1) implies that, for each , is uniformly bounded with respect to . Given that , we let be an integer such that . Then,
By (32),
Thus, we have
for each , where . Hence,
Now, we have that is uniformly bounded with respect to since
Since is increasing with respect to , Assertion follows from (32). Since is convex, we have
and Assertion ) also holds. Assertion is a direct consequence of Assertion . Assertion follows from Assertion . Furthermore, (34) and Assertion imply that the function is equicontinuous with respect to . By Sobolev’s theorem compactly for each . This means that, for any , is contained in a sequentially compact subset of which is independent of and . Thus, by the Ascoli-Arzela theorem, we obtain Assertion . Now, we have, for ,
the right hand side of which converges to as by (38) and Assertion (6). Now, we have Assertion (7). Assertions (2) and (7) imply Assertions (8) and (9). Letting in (34), we have
Thus, by Assertion (1) the left hand side is uniformly bounded with respect to and, hence, passing to a subsequence if necessary, converges weakly in , and by Assertion (6), the weak limit is . Then, by the lower semicontinuity of norm, we have
which implies Assertion (10).

Now, our purpose is to show that is a weak solution to (19). Proposition 9 implies that satisfies (i) and (ii) of Definition 7.

Since is a minimizer of , we have
Let us write, for , . Then, for each and for -a.e. ,
namely, for each ,
By Proposition 9 (7) we have that, for -a.e. ,
strongly in as . Let be a number such that (46) and (47) hold. We insert an arbitrary function in (46). Integrating it from to , we have the following by Proposition 9 (4), (7), Fatou’s lemma, and the lower semicontinuity:
For a while, we write for simplicity. First, we note the following identity:
Let be the integer such that . By (49), we have
By (42) and Proposition 9 (1), is uniformly bounded with respect to and . On the other hand, since ,
Hence, we have by (37)
and thus by (37) again
as . By Proposition 9 (1),
as . Since , , (47) implies that in . In particular, we have . Then, since , we have
Summing up, we have
It is not difficult to show that strongly in . Hence, by (48), we finally have
for each . By the convexity of , for each function with , we have . Thus, (57) implies that
It is easy to extend this inequality to all nonnegative functions . Hence, (iii) of Definition 7 holds for -a.e. .

4. Proof That Is a Contraction

Let , be functions in . In this section, we write
Then,
Let , be a solution to (19) with (3) and (4) for , , respectively. By (iii) of Definition 7,
Summing these, we have
Integrating from to , we have the following by (60) and by the fact that :
We further integrate this from to and write it . Then,
Here,
Thus,
These two terms are estimated as follows:
Summing up, we have
Hence, when , putting
we have
As , converges to . Thus, if is sufficiently small, . This means that the map from to is a contraction in . Hence, there is a fixed point and it is a solution to (10) with (3) and (4) in .

Uniqueness of a Local Solution. Let , be solutions to (10) with (3) and (4). Then, in the same calculus as before, we obtain
This implies the uniqueness.

Existence of a Time Global Solution. Suppose that is a solution to (10) with (3) and (4) in . First, we remark that
and the right hand side belongs to . Hence, we are able to solve (10) with (3) and (4) from . By the change of variable , (10) becomes
where
Hence, solving (10) with (4) and initial condition , , we obtain a function in which solves (10). For , put . Then, is a solution to (10) with (3) and (4) in . Repeating this process, we obtain a time global solution. Uniqueness of the local solution implies uniqueness of the global solution.

5. Uniform Estimates

Let be the small number presented in the previous section, and let be a solution to (10) with (3) and (4) in . As we see in the previous section, it is obtained as a fixed point of the map in , where is a solution to (19) with (3) and (4). The fixed point is obtained as in the following way. Let be an arbitrary element of and put . There is a constant which is determined by such that and . Thus, we have
and hence,
namely, is a Cauchy sequence in and it converges to a function . Since , we have that by letting . Thus, is the fixed point of .

Letting in (78), we have
By the lower semicontinuity, an energy inequality for a solution to (19) with (3) and (4) is obtained by letting in (32):
Since is increasing with respect to , we have . Hence,
Recall that is presented as in (33). By the proof of Proposition 9 (5), we have
Letting in (79), we finally have
Now, a solution to (10) is a solution to (19) for , the fixed point of . Hence, by (81), we have

By (83) and (84), there exists a constant such that
Now, by (83) and (85), we obtain uniform estimates for , , for the small .